Advanced Algebra（Abstract Part）
作者：任北上 主编
出版时间：2012年版
内容简介
高等代数是数学专业的一门重要的基础课程。它以矩阵、向量、线性空间和线性变换作为主要的研究对象，对培养学生的抽象思维能力、逻辑推理能力，以及数学专业的若干后续课程的学习都起着非常重要的作用。任北上编著的《Advanced Algebra（AbstractPart）》在每一章我们都选择了大批具有典型意义的例题，帮助学生举一反三，触类旁通，其中有一些就是本课程的重要结论。通过例题的学习，学生不仅可以更容易地理解抽象的数学概念和内容，疏通各知识链条环环相扣的彼此联系，而且更便于加深对课堂内容的吸纳和消化，从中掌握本课程的数学思想和数学方法。
目录
Chapter 1Linear Spaces(线性空间)(1)
1.1Basic Concept（基本概念）(1)
1.1.1Integers (整数)(1)
1.1.2Mappings （映射）(3)
1.1.3Equivalence Relation (等价关系)(6)
1.1.4Exercises and Supplementary Exercises（习题及补充练习）(7)
1.2Definition, Examples and Simple Properties of LinearSpaces(线性空间的定义、
例子和简单性质)(8)
1.2.1Definition and Examples of Linear Space (线性空间的定义和例子)(8)
1.2.2Properties of Iinear Space （线性空间的性质）(10)
1.2.3Exercises and Supplementary Exercises（习题及补充练习）(11)
1.3Dimension, Basis and Coordinates (维数、基与坐标)(14)
1.3.1Linear Combination and Linear Dependence (线性组合及线性相关)(14)
1.3.2Basis and Dimension of Linear Space（线性空间的基与维数）(15)
1.3.3Coordinate of a Vector with Respect to theBasis（向量关于基的坐标）(17)
1.3.4Exercises and Supplementary Exercises（习题及补充练习） (18)
1.4Basis Change and Coordinate Transformations（基变换与坐标变换）(20)
1.4.1Basis Change (基变换)(20)
1.4.2Coordinate Transformations（坐标变换）(22)
1.4.3The Properties of the Transition Matrix（过渡矩阵的性质）(23)
1.4.4Exercises and Supplementary Exercises（习题及补充练习）(27)
1.5Linear Subspaces (线性子空间)(29)
1.5.1Definition and Examples of Linear Subspace(线性子空间的定义和例子)(29)
1.5.2Linear Subspaces Generated by a Set of Vectors(由向量组生成的线性子空间)(31)
1.5.3Intersection Subspace and Sum Subspace （交子空间与和子空间）(33)
1.5.4Direct Sum of Subspaces （子空间的直和）(36)
1.5.5Exercises and Supplementary Exercises（习题及补充练习）(39)
1.6Isomorphism of Linear Spaces (线性空间的同构)(40)
1.6.1Definition and Simple Properties of Isomorphism of LinearSpaces (线性空间同构的定义
和简单性质)(40)
1.6.2The Application of Isomorphism of Linear Spaces(线性空间同构的应用)(43)
1.6.3Exercises（习题）(46)
1.7※Factor Spaces (商空间)(46)
1.7.1Properties of Cosets (陪集的性质)(46)
1.7.2Factor Space (商空间)(48)
Test for Chapter 1 (第1章测试卷)(49)
Biography of A. L. Cauchy(53)
Chapter 2Linear Transformations (线性变换)(54)
2.1Definition and Operation of LinearTransformation（线性变换的定义和运算）(54)
2.1.1Definition，Examples and Basic Properties of LinearTransformation (线性变换的定义、范例及
基本性质)(54)
2.1.2Operation of Linear Transformations （线性变换的运算）(56)
2.1.3The Image and Kernel of a LinearTransformation（线性变换的像与核）(59)
2.1.4Exercises and Supplementary Exercises(63)
2.2The Matrix of a Linear Transformation (线性变换的矩阵)(65)
2.2.1Matrix of a Linear Transformation with Respect to the Basis(线性变换关于基的矩阵)(65)
2.2.2The Correspondence Relation Between the Linear Transformationand the Matrix （线性变换
与矩阵之间的对应关系）(67)
2.2.3The Relationship between the Coordinates of a Vector and ItsImage（向量与它的像的坐标
之间的关系）(71)
2.2.4Exercises and Supplementary Exercises(77)
2.3Invariant Subspaces (不变子空间)(80)
2.3.1Definition and Examples of Invariant Subspace(不变子空间的定义和例子)(81)
2.3.2The Relationship between the Invariant Subspace and SimplifiedMatrix (不变子空间与化
简矩阵的关系)(82)
2.3.3Exercises and Supplementary Exercises(85)
2.4Eigenvalues and Eigenvectors (特征值及特征向量)(87)
2.4.1Concept of Eigenvalues and Eigenvectors of a LinearTransformation (线性变换的特征值和特征
向量的概念)(87)
2.4.2Method for Finding the Eigenvalues and Eigenvectors(特征值和特征向量的求法)(89)
2.4.3The Eigenvectors of A and A?subspaces(A的特征向量及A?子空间)(93)
2.4.4Exercises and Supplementary Exercises(96)
Test for Chapter 2 (第2章测试卷)(98)
Biography of A.Cayley(102)
Chapter 3Euclidean Spaces(欧几里得空间)(103)
3.1Concept of Euclidean Spaces (欧几里得空间的概念）(103)
3.1.1Definition and Examples of Euclidean Spaces(欧几里得空间的定义及实例)(103)
3.1.2Basic Properties of Euclidean Spaces (欧几里得空间的基本性质）(105)
3.1.3Exercises and Supplementary Exercises(112)
3.2Orthonormal Bases (标准正交基)(114)
3.2.1Orthogonal Set, Orthonormal Set, Orthogonal Basis andOrthonormal basis(正交组，标准
正交组，正交基及标准正交基）(114)
3.2.2Existence of the Orthonormal Basis and SchmidtOrthogonalization Procees(标准正交基的
存在性与施密特正交化过程)(120)
3.2.3The Isomorphism of Euclidean Spaces(欧几里得空间的同构)(123)
3.2.4Exercises and Supplementary Exercises(124)
3.3Orthogonal and Symmetric LinearTransformations(正交线性变换及对称线性变换)
-126
3.3.1Orthogonal Linear Transformations (正交线性变换)(127)
3.3.2Symmetric Linear Transformations (对称线性变换)(130)
3.3.3Exercises and Supplementary Exercises(131)
3.4Orthogonal Complement of Subspaces(子空间的正交补) (134)
3.4.1Definition and Properties of the Orthogonal Complement ofSubspaces（子空间的正交补的
定义和性质）(134)
3.4.2Exercises and Supplementary Exercises(136)
3.5※Conjugate Linear Transformations and UnitarySpaces（共轭线性变换及酉空间）
-138
3.5.1Conjugate Linear Transformations (共轭线性变换)(138)
3.5.2Unitary Spaces （酉空间）(140)
3.5.3Exercises and Supplementary Exercises(147)
Test for Chapter 3 (第3章测试卷)(148)
Biography of Euclid(152)
Chapter 4Matrices Similar to Diagonal Matrices(矩阵相似于对角形)(153)
4.1Diagonalization of Matrices (矩阵的对角化）(153)
4.1.1Eigenvalues, Eigenvectors and Characteristic Polynomials of aMatrix(矩阵的特征值、特征
向量及特征多项式)(153)
4.1.2Concept of Diagonalization for Matrices (矩阵对角化的概念)(158)
4.1.3The Relationship between the Diagonalization of A andA(矩阵A与线性变换A的对角化之
间的关系)(162)
4.1.4Exercises and Supplementary Exercises(165)
4.2Diagonalization of Real Symmetric Matrices and SymmetricTransformations
（实对称矩阵及对称变换的对角化）(167)
4.2.1Basic Properties and Theorems（基本性质和基本定理）(167)
4.2.2Diagonalization of Real Symmetric Matrices and SymmetricTransformations（实对称矩阵及
对称变换的对角化）(169)
4.2.3Examples （范例）(173)
4.2.4Exercises and Supplementary Exercises(174)
4.3Cayley?Hamilton Theorem and Minimum Polynomial（凯莱?哈密尔顿定理及最
小多项式）(176)
4.3.1Cayley ? Hamilton Theorem (凯莱?哈密尔顿定理)(176)
4.3.2Minimum Polynomials （最小多项式）(178)
4.3.3Exercises and Supplementary Exercises(183)
Test for Chapter 4 (第4章测试卷)(185)
Biography of C. Hermite(188)
Chapter 5Jordan Canonical Form ofMatrices(矩阵的若当标准形)(190)
5.1Invariant Factor, Determinant Division and Condition forMatrices to be Similar
（不变因子、行列式因子及矩阵相似的条件）(190)
5.1.1Necessary and Sufficient Condition for Two Matrices to beSimilar（两个矩阵相似的充分必
要条件）(190)
5.1.2Invariant Factor, Determinant Division and Canonical form ofλ?Matrices(不变因子、行列式
因子及λ?矩阵的标准形)(194)
5.1.3Exercises and Supplementary Exercises(199)
5.2Elementary Divisor and Jordan Canonical Forms(初等因子及若当标准形)(201)
5.2.1Necessary and Sufficient Condition for Two λ?Matrices to beEquivalent(两个α?矩阵等价的
充分必要条件)(201)
5.2.2Basic Properties and Application of Jordan CanonicalForms(若当标准形的基本性质及应用)
-206
5.2.3※Rational Canonical Forms of the Matrices(矩阵的有理标准形)(211)
5.2.4Exercises and Supplementary Exercises(213)
Test for Chapter 5 (第5章测试卷)(216)
Biography of C. Jordan(219)
Chapter 6Quadratic Forms (二次型)(221)
6.1Standard Forms of General Quadratic Forms(二次型的标准形）(221)
6.1.1The Matrix Expression of Quadratic Forms and LinearSubstitution of Variables (二次型的矩阵表示以及变量的线性代换)(222)
6.1.2Equivalence of Quadratic Forms and Congruence of Matrices(二次型的等价及矩阵的合同)
-224
6.1.3Sum of Squares and Standard Forms of Quadratic Forms(二次型的平方和与标准形)(224)
6.1.4Exercises and Supplementary Exercises(229)
6.2Properties and Classification of Real QuadraticForms(实二次型的性质及分类）
-231
6.2.1Standard Forms of Real Quadratic Forms(实二次型的标准形)(231)
6.2.2Classification of Real Quadratic Forms (实二次型的分类)(235)
6.2.3Another Method for Determining of the Positive Definitenessand the Negative Definiteness
of a Real Quadratic Form (确定实二次型的正定性和负定性的其他方法)(237)
6.2.4Exercises and Supplementary Exercises(240)
Test for Chapter 6 (第6章测试卷)(241)
Biography of P.S.Laplace(245)
Chapter 7Bilinear Functions (双线性函数)(247)
7.1Linear Mappings （线性映射）(247)
7.1.1Definition, Examples and Basic Properties of Linear Mapping(线性映射的定义、范例和基本性质)
-247
7.1.2The Restriction and Extension of a Linear Mapping(线性映射的限制及扩张)(252)
7.1.3The Universal Properties of a Linear Mapping(线性映射的泛性质)(253)
7.1.4Direct Sum of Linear Spaces and Linear Mappings(线性空间和线性映射的直和)(256)
7.1.5Exercises and Supplementary Exercises(258)
7.2Bilinear Functions(双线性函数)(260)
7.2.1Linear Functions (线性函数)(260)
7.2.2Bilinear Functions (双线性函数)(260)
7.2.3Exercises and Supplementary Exercises(264)
7.3Dual Spaces (对偶空间）(266)
7.3.1Dual Spaces (对偶空间)(266)
7.3.2Dual Mappings (对偶映射)(269)
7.3.3Exercises and Supplementary Exercises(272)
Test for Chapter 7 (第7章测试卷)(274)
Biography of L.Kronecker(278)
Index(中?英文名词索引)(279)
Bibliography(283)AOE
